arXiv subjects

Fabian Waleffe

Publications and source records attributed to Fabian Waleffe.

8 recordsLinked to original sources

arXiv2015

Optimal Heat Transport in Rayleigh-Bénard Convection

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-Bénard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Ra\sim 10^9$. Power law scalings of $Nu\sim Ra^γ$ are observed with $γ\approx 0.31$, where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $Pr$ is found to be extremely weak. On the other hand, the presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $Pr \lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid which spawns left/right horizontal arms near the top of the channel, leading to downdrafts adjacent to the central updraft. For $Pr > 7$ at high-enough Ra, the optimal structure is a single updraft absent significant horizontal structure, and characterized by the larger maximal wavenumber.

physics.flu-dyn↗

arXiv2015

Heat Transport by Coherent Rayleigh-Bénard Convection

Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number $Ra \approx 1708$ has been calculated up to $Ra\approx 5. 10^6$ and shows heat flux $Nu \sim 0.143\, Ra^{0.28}$ with a delicate spiral structure in the temperature field. Another solution that maximizes $Nu$ over the horizontal wavenumber has been calculated up to $Ra=10^9$ and its heat flux scales as $Nu \sim 0.115\, Ra^{0.31}$ for $10^7 < Ra \le 10^9$, quite similar to 3D turbulent data. The latter is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as $0.133 \, Ra^{0.217}$ in that range. That optimum solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength and spacing of elemental plumes.

physics.flu-dyn↗

arXiv2007

Gegenbauer tau methods with and without spurious eigenvalues

It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative and distinct eigenvalues, as is the case for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and `inviscid' Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results show that the results are sharp: positive or complex eigenvalues arise outside of this class. The widely used modified tau approach is proved to be equivalent to the Galerkin method.

math.NA↗

arXiv2007

Lower branch coherent states in shear flows: transition and control

Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of O(1) streaks, $O(R^{-1})$ streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number $R$. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggests that the lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

physics.flu-dyn↗

arXiv2006

Spectrum of the Jacobi tau approximation for the second derivative operator

It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer Tau method where the range of parameters included in the theorems can be extended and characteristic polynomials given by successive order approximations interlace. This includes the common Chebyshev and Legendre, Tau and Galerkin methods. The characteristic polynomials for the Gegenbauer Tau method are shown to obey three term recurrences plus a constant term which vanishes for the Legendre Tau and Galerkin cases. These recurrences are equivalent to a tridiagonal plus one row matrix structure. The spectral integration formulation of the Gegenbauer Tau method is shown to lead directly to that fundamental and well-conditioned tridiagonal plus one row matrix structure. A Matlab code is provided.

math.NA↗

arXiv2004

On some dyadic models of the Euler equations

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the $H^{3/2+ε}$ Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in $H^α$, for $α\ge 1/2$, but its dyadic restriction is even more singular, exhibiting blow-up for any $α> 0$. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in $H^{3/2+ε}$. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the $H^α$ norm, with $α> 0$, is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

math.AP↗

arXiv2004

Remarks on a quasi-linear model of the Navier-Stokes Equations

Dinaburg and Sinai recently proposed a quasi-linear model of the Navier-Stokes equations. Their model assumes that nonlocal interactions in Fourier space are dominant, contrary to the Kolmogorov turbulence phenomenology where local interactions prevail. Their equation corresponds to the linear evolution of small scales on a background field with uniform gradient, but the latter is defined as the linear superposition of all the small scale gradients at the origin. This is not self-consistent.

math.AP↗

arXiv2002

Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows

Nontrivial steady flows have recently been found that capture the main structures of the turbulent buffer layer. We study the effects of polymer addition on these "exact coherent states" (ECS) in plane Couette flow. Despite the simplicity of the ECS flows, these effects closely mirror those observed experimentally: Structures shift to larger length scales, wall-normal fluctuations are suppressed while streamwise ones are enhanced, and drag is reduced. The mechanism underlying these effects is elucidated. These results suggest that the ECS are closely related to buffer layer turbulence.

physics.flu-dyn↗